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Extrasolar Planets and Stellar
Oscillations in K Giant Stars
Notes can be downloaded from
www.tls-tautenburg.de→Teaching
Spectral Class
O
B
A
F
G
K
M
-10
Supergiants
-5
1.000.000
10.000
0
Giants
100
+5
1
+10
0.01
White
Dwarfs
0.0001
+15
+20
20000
14000
10000
7000
5000
Effective Temparature
3500
2500
Why the interest in K giants for exoplanets and
asteroseismology?
Evolved A-F stars
K giants occupy a „messy“
region of the H-R diagram
Progenitors are higher mass
stars
The story begins:
Smith et al. 1989 found a 1.89 d period in Arcturus
1989 Walker et al. Found that RV variations are common
among K giant stars
These are all
IAU radial
velocity
standard stars
!!!
First, planets around K giants stars…
1990-1993 Hatzes & Cochran surveyed 12 K giants with
precise radial velocity measurements
Footnote: Period Analysis
Lomb-Scargle Periodogram:
Px(w) =
1
2
[ S X cos w(t –t)]
j
j
S
2
j
1
+
2
2
Xj cos w(tj–t)
j
tan(2wt) =
[ S X sin w(t –t) ]
j
j
j
S
Xj sin2 w(tj–t)
(Ssin
2wtj)/ (Scos 2wtj)
j
j
Power is a measure of the statistical significance of that
frequency (period):
False alarm probability ≈ 1 – (1–e–P)N = probability that noise
can create the signal
N = number of indepedent frequencies ≈ number of data points
2
If a signal is present, for less noise (or more data) the power of
the Scargle periodogram increases. This is not true with
Fourier transform -> power is the related to the amplitude of
the signal.
Many showed RV
variations with periods of
200-600 days
p Her has a 613 day
period in the RV
variations
But what are the
variations due to?
The nature of the long period variations in K giants
Three possible hypothesis:
1. Pulsations (radial or non-radial)
2. Spots (rotational modulation)
3. Sub-stellar companions
What about radial pulsations?
Pulsation Constant for radial pulsations:
Q=
M 0.5
P(
)
M‫סּ‬
R –1.5
(R ) =
‫סּ‬
P
(
r 0.5
)
r‫סּ‬
For the sun:
Period of Fundamental (F) = 63 minutes = 0.033 days (using
extrapolated formula for Cepheids)
Q = 0.033
Footnote:
The fundamental radial mode is related to the dynamical
timescale:
d2R
dt2
GM
=
R2
The dynamical timescale is the time it takes a star to
collapse if you turn off gravity
Approximate:
R
t2
≈GrR
t = (Gr)–0.5
For the sun t = 54 minutes
r is the mean
density
What about radial pulsations?
K Giant: M ~ 2 M‫ סּ‬, R ~ 20 R‫סּ‬
Period of Fundamental (F) = 2.5 days
Q = 0.039
Period of first harmonic (1H) = 1.8 day
→ Observed periods too long
What about radial pulsations?
Alternatively, let‘s calculate the change in radius
V = Vo sin (2pt/P),
p/2
DR =2 ∫ Vo sin (2pt/P) =
VoP
p
0
b Gem: P = 590 days, Vo = 40 m/s, R = 9 R‫סּ‬
DR ≈ 0.9 R‫סּ‬
Brightness ~ R2
Dm = 0.2 mag, not supported by Hipparcos photometry
What about non-radial pulsations?
p-mode oscillations, Period < Fundamental mode
Periods should be a few days → not p-modes
g-mode oscillations, Period > Fundamental mode
So why can‘ t these be g-modes?
Hint: Giant stars have a very large, and deep convection zone
Recall gravity modes and the Brunt–Väisälä Frequency
The buoyancy frequency of an oscillating blob:
N2
=g
G1 = (
(
1 r dP
G1 P dr
r dP
P dr
)ad
–
dr
dr
)
First adiabatic exponent
g is local acceleration of gravity
r is density
P is pressure
Where does this
come from?
Brunt Väisälä Frequency
r
DT
Dr
r*
Change in density of surroundings:
r = r0 +
Dr
( )
Dr
Change in density due to adiabatic
expansion of blob:
r* = r0 +
r0
dr
dr
(
dr dP
dP dr
(
1
G1
r0
r* = r0 +
r
P
) Dr
dP
dr
) Dr
Brunt Väisälä Frequency
r
Difference in density between blob
and surroundings :
DT
Dr
r*
Dr = r – r*
Dr
=
(
1
G1
r
P
dP
dr
–
dr
dr
1
1 dr
dP
–
= –r
r dr
G1 P dr
(
r0
r0
Recall F = –kx
) Dr
) Dr
Buoyancy force fb = – gDr dr
→ w2 = k/m
This is just a harmonic oscillator with w2 = N2
Brunt Väisälä Frequency
However if r* < r, the blob is less dense than its
surroundings, buoyancy force will cause it to continue to rise
Criterion for onset of convection:
(
1
G1
r
P
dP
dr
)
dr
<
dr
In convection zone buoyancy is a destabilizing force,
gravity is unable to act as a restoring force → long
period RV variations in K giants cannot be g modes
What about rotation?
Radius of K giant ≈ 10 R‫סּ‬
Rotation of K giant ≈ 1-2 km/s
Prot ≈ 2pR/vrot
Prot ≈ 250–500 days
Its possible!
Spots can
cause RV
variations
Rotation (and pulsations) should be accompanied
by other forms of variability
Planets on the other hand:
1. Have long lived and coherent RV variations
2.
No chromospheric activity variations with RV period
3.
No photometric variations with the RV period
4.
No spectral line shape variations with the RV period
Case Study: b Gem
CFHT
McDonald 2.1m
McDonald 2.7m
TLS
Ca II H & K core emission is a measure of magnetic activity:
Active star
Inactive star
Ca II emission variations
Hipparcos Photometry
Test 2: Bisector velocity
From Gray (homepage)
Spectral line shape variations
The Planet around b Gem
Period
RV Amplitude
e
a
Msin i
590.5 ± 0.9 d
40.1 ± 1.8 m/s
0.01 ± 0.064
1.9 AU
2.9 MJupiter
The Star
M = 1.7 Msun
[Fe/H] = –0.07
P = 1.5 yrs
Frink et al. 2002
M = 9 MJ
Setiawan et al. 2005
P = 711 d
Msini = 8 MJ
Setiawan et al. 2002:
P = 345 d
e = 0.68
M sini = 3.7 MJ
a Tau
Hatzes & Cochran 1998
a Tau has line
profile variations,
but with the wrong
period
The Planet around a Tau
Period
RV Amplitude
e
a
Msin i
653.8 ± 1.1 d
133 ± 11 m/s
0.02 ± 0.08
2.0
10.6 MJupiter
The Star
M = 2.5 Msun
[Fe/H] = –0.34
g Dra
The Planet around g Dra?
Period
RV Amplitude
e
a
Msin i
712 ± 2.3 d
134 ± 9.9 m/s
0.27 ± 0.05
2.4
13 MJupiter
The Star
M = 2.9 Msun
[Fe/H] = –0.14
The evidence supports that the long period RV variations in many K
giants are due to planets…so what?
Setiawan et al. 2005
B1I V
F0 V
G2 V
Planets around massive K giant stars
g Dra
2.9
13
2.4
712
0.27
–0.14
a Tau
2.5
10.6
2.0
654
0.02
–0.34
Period
Characteristics:
1. Supermassive planets: 3-11 MJupiter
Theory: More massive stars have more massive disks
2. Many are metal poor
Theory: Massive disks can form planets in spite of low
metallicity
3. Orbital radii ≈ 2 AU
Theory: Planets in metal poor disks do not migrate
because they take so long to form.
And now for the stellar oscillations…
Hatzes & Cochran 1994
Short period
variations in
Arcturus
n = 1 (1H)
n = 0 (F)
n
0
1
2
F
1H
2H
a Ari
Alias
n≈3 overtone radial mode
g Dra
g Dra : June 1992
g Dra : June 2005
g Dra
Photometry of a UMa with WIRE guide camera (Buzasi et al. 2000)
Radial modes n =
0
1
2
3
Conclusion: most (all?) K giant stars pulsate in
the radial and low-overtone modes.
So what?
HD 13189
P = 471 d
Msini = 14 MJ
M* = 3.5 s.m.
P = 4.8 days
For M = 3.5 M‫סּ‬
R = 38 R
F = 4.8 d
2H = 2.7 d
→ oscillations can be
used to get the stellar
mass
HD 13189 short
period variations
P = 2.4 days
Current work on K giants
1. TLS survey of 62 K giants (Döllinger Ph.D.)
2. Multi-site campaigns planned (GLONET)
3. MOST campaign on b Oph and b Gem
4. CoRoT additional science program (150 days of
photometry)
5. Lots of theoretical work to model pulsations needs
to be done
Döllinger Ph.D. work: 62 K giants surveyed from TLS
≈ 10% show long period variations that may be due to planetary
companions
Aldebaran with MOST
Intensity
5.7 days
Time (days)
Summary
• K giant (IAU radial velocity standards) are RV
variable stars!
• Multi-periodic on two time scales: 200-600 days and
0.25 – 8 days
• Long period variations are most likely due to giant
planets around stars with Mstar > 1 M‫סּ‬
• Short period variations are due to radial pulsations in
the fundamental and overtone modes
• Pulsations can be used to get funamental parameters of
star